Stability under $\Gamma$-Convergence of Least Energy Solutions for Semilinear Problems in the Whole $R^N$

Abstract

We study the homogenization of semilinear elliptic equations in divergence form with discontinuous oscillating coefficients in the whole $\mathbb{R}^N$. As is well known, the homogenization process in a classical framework is concerned with the study of asymptotic behavior of solutions $u_{\epsilon}$ of boundary value problems when the period $\epsilon>0$ of the coefficients is small. By extending some of the classical homogenization results for quasi-linear elliptic equations to unbounded domains and, making use of various variational techniques, we shall establish some stability results under $\Gamma$-convergence of least energy solutions for such boundary value problems.